This makes sense in a microcosm, but it took 6 days after the last atomic attack for them to surrender (after having been hit twice in 3 days).
The bombing of Nagasaki was reported to the Supreme Council at 11 AM on 9 August. 15 hours later, at 2 AM on 10 August, Hirohito spoke for the first time: he agreed that Japan would surrender with no conditions except a "guarantee of the Emperor's position". IOW, Japan decided to surrender then. Japan telegraphed its acceptance of the Potsdam Declaration that morning, with that one reservation.
What changed in those 6 days? The Soviets dropped their non-aggression pact with Japan...
Early on 9 August,
before the bombing of Nagasaki. This had no effect on the Supreme Council.
... forcing them to the realization that a negotiated peace with the WAllies via the Soviets was not going to happen.
That was already known, but did not sway the hardliners.
The timeline is:
26 July - Allies issue Potsdam Declaration, setting terms for Japanese surrender
27 July - Japan rejects the Declaration
6 August - bombing of Hiroshima; no change in Japanese position
9 August - 4 AM: Soviet attack; no change in Japanese position; 11 AM: bombing of Nagasaki; Supreme Council splits 3-3 on surrender
10 August - 2 AM: Hirohito speaks out; 10 AM: Japan accepts Declaration with one reservation
12 August - Allies reject Japanese offer
13 August - Allies drop leaflets describing Japanese surrender offer
14 August - Allies resume bombing Japan; Japan accepts Potsdam Declaration without reservations; Hirohito records surrender speech
15 August - Surrender speech broadcast
I think the record is clear. Japan decided to surrender on 10 August, after the bombing of Nagasaki. Up to that point, Japanese leaders rejected any call for surrender. After Nagasaki, the hardliners proposed surrender but on Japanese terms (no occupation, etc), while the moderates wanted only the guarantee of the Emperor - splitting 3-3. Several hours later, Hirohito spoke, supporting the moderates, and the council agreed. The Japanese tried to quibble over the Emperor's status for a few days, but they had accepted surrender.
One should note that on 14 August, as junior hardliners plotted a coup d'état to prevent the surrender announcement, several senior officers signed an agreement to follow the Emperor's order. This group included two of the three hardliners on the Supreme Council.